The above, despite being a natural question, has only been studied for very specific instantiations of \(\psi \)  such as Max-\(k\) -Vertex Cover [1]}, Max-Cut [2]}. Furthermore, tight lower bounds are known for even fewer settings such as the almost satisfiable regimes of SmallSetExpansion, Max-Bisection, BalancedSeparator [3]}, and as such, a finer understanding of \(\alpha _{\le \mu }(\cdot )\)  is absent even for natural problems such as \({\sf D}k{\sf S}\) .
